Should the input to the negative log likelihood loss function be probabilities?
This seems pretty reasonable to me. You can optimize for any function that is proportionate to the negative log-likelihood. Conventionally, we assume that the likelihood of piece of data under a linear model is proportionate to some sort of gaussian function of the difference between the predicted value and the observed value. If you're a Bayesian you'd say this is a probability. If you're a hardcore frequentist, you might quibble about that, but it's still a number between 0 and 1.

If you take the log of this likelihood function however, you'll get a quadratic function of the difference, with some scalers that you don't need to worry about. So you ought to minimize:

$$\sum_y (y-\hat{y})^2$$

This is not a probability, but since it is proportionate to the log of the original likelihood function (which was, in some sense), minimizing it will also minimize the original function.

Hope that helps!